Peter (not Paul) Preston had a poorly-written piece about crime in Barcelona with several inaccuracies on the Guardian's website. The piece is so poor that the Guardian has taken it off its Comment Is Free archive, but you can still access it on Google.
However, Preston is absolutely right about one thing: Street crime is rampant in Barcelona, and all tourists are targets. You're not likely to get hurt, but it is likely that somebody will try to pick your pocket or snatch your bag. Watch for groups of Moroccan teenagers wearing expensive sports clothing and for Bosnian women pickpockets on the tourist bus.
You should be OK if you take proper precautions and keep an eye out. Don't carry anything valuable on you. Don't let your possessions get out of your sight at a restaurant or bar. Don't get sloppy drunk, don't try to buy drugs, don't go down any dark streets, stay out of the Raval at night, and it's best if you go out accompanied. Consider staying at a hotel outside the Old City, in the Eixample or Sant Gervasi, which are considerably safer areas. Particularly dangerous tourist areas: Public transport, the Ramblas and Barrio Gotico, the Sagrada Familia, La Pedrera, the Picasso Museum, and the Parque Guell. Whatever you do, stay away from the three-card monte dealers on the Ramblas. Not only is the game a scam, of course, but pickpockets act in the crowd watching it. Also, if you are driving, and somebody (not a marked police car) indicates that you should pull over, don't do it until you get to a place where there are some other people.